Plasma-induced neutrino helicity flip in a supernova core and a constraint on the Dirac neutrino...

ISSN 1063�7761, Journal of Experimental and Theoretical Physics, 2010, Vol. 111, No. 1, pp. 70–81. © Pleiades Publishing, Inc., 2010.Original Russian Text © A.V. Kuznetsov, N.V. Mikheev, A.A. Okrugin, 2010, published in Zhurnal Éksperimental’noі i Teoreticheskoі Fiziki, 2010, Vol. 138, No. 1, pp. 80–92.

70

1. INTRODUCTION

A nonzero neutrino magnetic moment opens upthe possibilities for the occurrence of various processeswith helicity flip in which the left�handed neutrinosproduced inside a star become right�handed ones, i.e.,sterile with respect to weak interactions. This can beimportant, for example, when the energy losses bystars are taken into account. In the minimallyextended standard model with massive neutrinos, theneutrino magnetic moment was obtained in [1, 2] inthe form

(1)

where GF is the Fermi coupling constant, mν is theneutrino mass, μB = e/2me is the Bohr magneton, and

me is the electron mass.1 Given the well�known con�straints on the neutrino masses, the quantity (1) is neg�ligible. On the other hand, nontrivial extensions of thestandard model (such as the left–right symmetry) canlead to more significant neutrino magnetic moments.

The neutrino helicity flip mechanism, νL νR,was first applied in an attempt to solve the problem ofsolar neutrinos. Two different scenarios were analyzed.The first scenario based on the rotation of the neutrinomagnetic moment in the magnetic field of a star wasconsidered in [3–5]. In the second scenario, the neu�trinos change their helicity through the electromag�netic interaction of their magnetic moment withplasma [6, 7]. A fuller list of works can be found, forexample, in [8]. In both approaches, the effect for μν <

1 We use a natural system of units with c = � = kB = 1; e > 0 is theelementary charge.

μν

SM( ) 3eGFmν

8π2 2�� 3.20 10 19– mν

1 eV��⎝ ⎠⎛ ⎞ μB,×= =

10–10μB was shown to be too small to affect signifi�cantly the solution of the problem of solar neutrinos.

Even more stringent constraints on μν originatefrom the physics of other stars. For example, the non�standard neutrino losses could be more pronounced inthe cores of low�mass red giants, where the plasma isdenser than that at the solar center approximately byfour orders of magnitude, which causes the heliumignition to be delayed. Accordingly, the following limitwas obtained [9, 10]:

(2)

An independent constraint on the neutrino mag�netic moment was also obtained by analyzing the evo�lution of the early Universe [11, 12]:

(3)

where the neutrino collisions with helicity flips couldincrease noticeably the number of sterile Dirac com�ponents at the epoch before neutrino decoupling.This, in turn, would cause the effective number ofthermally excited degrees of freedom of the neutrinosto double and the expansion rate of the Universe toincrease, which would lead to a helium overabun�dance.

Interest in possible astrophysical and cosmologicalmanifestations of the neutrino magnetic momentstimulated experiments on its measurement in labora�tory conditions. The best constraint was obtained inthe recent GEMMA experiment to study the scatter�ing of antineutrinos by electrons carried out at theKalinin nuclear power station by the collaboration ofthe Institute of Theoretical and Experimental Physics(Moscow) and the Joint Institute for Nuclear

μν 0.3 10 11– μB.×<

μν 6.2 10 11– μB,×<

Plasma�Induced Neutrino Helicity Flip in a Supernova Core and a Constraint on the Dirac Neutrino Magnetic Moment

A. V. Kuznetsov, N. V. Mikheev, and A. A. Okrugin*Yaroslavl State University, Sovietskaya ul. 14, Yaroslavl, 150000 Russia

*e�mail: [email protected] January 16, 2010

Abstract—We investigate the neutrino helicity flip under supernova core conditions, where the left�handedneutrinos being produced can be converted into right�handed neutrinos sterile with respect to weak interac�tions owing to the interaction of the magnetic moments with plasma electrons and protons. In calculating theprobability for the conversion neutrino scattering by plasma components, we take into account the polariza�tion effects attributable to both electrons and protons in the photon propagator. Based on realistic modelswith radial distributions and time evolution of physical parameters in a supernova core, we have obtainedupper limits on the Dirac neutrino magnetic moment averaged over flavors and time from the condition thatthe influence of the right�handed neutrino emission on the total cooling time scale should be limited.

DOI: 10.1134/S1063776110070071

NUCLEI, PARTICLES, FIELDS,GRAVITATION, AND ASTROPHYSICS

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 111 No. 1 2010

PLASMA�INDUCED NEUTRINO HELICITY FLIP IN A SUPERNOVA CORE 71

Research (Dubna). The upper bound for the neutrinomagnetic moment was [13]

(4)

This result improved the previous laboratory con�straint [14] by more than a factor of 2.

Noticeable impetus for the growth of interest in theneutrino magnetic moment was the momentous eventof the supernova SN1987A explosion. The huge neu�trino fluxes during a supernova explosion are known tobasically determine the energetics of the process,implying that such a microscopic neutrino character�istic as the magnetic moment can have a decisive influ�ence on the macroscopic properties of these astro�physical events. The left�handed neutrinos trappedinside the supernova core during its collapse can beconverted into right�handed ones. This process is pos�sible owing to the interaction of the Dirac neutrinomagnetic moment with a virtual plasmon γ* that canbe both produced and absorbed:

νL νR + γ*, νL + γ* νR. (5)Leaving the core, these sterile neutrinos might notleave a sufficient energy needed to explain theobserved supernova neutrino luminosity. Hence anupper bound can be placed on the neutrino magneticmoment.

This question was studied in various aspects by theauthors of [15–19]. For example, the neutrino helicityflip through the scattering processes inside the super�nova core, νLe– νRe– and νLp νRp, immedi�ately after its collapse was considered in [17]. Taking

~ 1053 erg/s as an upper limit for the right�handed

neutrino luminosity, the authors of [17] obtained thefollowing upper bound on the neutrino magneticmoment:

(6)

However, in [17] the significant (in the case underconsideration) plasma polarization effects were notincluded in the photon propagator. Instead, the pho�ton dispersion was taken into account phenomenolog�ically by including a specially chosen thermal photonmass. This question was investigated in detail in[20, 21], where the formalism of the thermal field the�ory was used to allow for the influence of a hot denseastrophysical plasma on the photon propagator. Theupper bound for the neutrino magnetic momentturned out to be half [20, 21] that in [17]:

(7)

Considering the intermediate results of analytical cal�culations presented in [20, 21], we can see that theauthors took into account only the plasma electroncontribution, while they neglected the proton contri�bution. Note that in [17] the electron and proton con�tributions to the neutrino helicity flip were consideredas contributions of the same order of magnitude. Itshould also be noted that the improvement in con�

μν 3.2 10 11– μB.×<

LνR

μν 0.2–0.8( ) 1011– μB.×<

μν 0.1–0.4( ) 10 11– μB.×<

straint (7) with respect to constraint (6) was based inpart on the overestimation of the supernova core vol�ume in [20, 21] by a factor of 2 compared to that in[17], while the density was taken to be the same, ρc ≈8 × 1014 g/cm3. This means that the core mass in[20, 21] was ~3M� (M� is the solar mass), which isapproximately twice the mass of the supernova rem�nant considered to be typical.

The processes (5) with neutrino helicity conversionattributable to the interaction with both plasma elec�trons and protons through a virtual plasmon and withallowance made for the plasma polarization effects onthe photon propagator were analyzed in detail in [22].In particular, the plasma proton contribution turnedout to be not only significant but even dominant.

However, all of the previous studies [15–22] werebased on highly simplified models of the supernovacore as a homogeneous sphere with averaged physi�cal parameters. Moreover, according to presentviews, the parameters more likely appear overesti�mated than typical.

The goal of this paper is to estimate the Dirac neu�trino magnetic moment based on a constraint on thecontribution from the emission of neutrinos νR to thesupernova core luminosity by a more reliable methodthan that in previous papers using the radial distribu�tions and time evolution of physical parameters fromseveral more realistic supernova core models.

We consider only Dirac neutrinos, because, in thiscase, the interaction of the neutrino magnetic moment(both diagonal and nondiagonal components) withphotons converts the active left�handed neutrinos intoright�handed ones that are sterile with respect to weakinteractions. The case of Majorana neutrinos in thisaspect is of no interest, because the right�handedantineutrino states being produced are not sterile.

The amplitude of the helicity flip through the scat�tering by plasma components is calculated in Section 2.A general expression for the creation probability ofright�handed neutrinos with a fixed energy is derived.In Section 3, we estimate the core luminosity withrespect to the emission of neutrinos νR and obtain anupper limit on the neutrino magnetic moment by tak�ing into account the radial distributions and time evo�lution of physical parameters.

2. THE INTERACTION OF NEUTRINOS WITH AN ASTROPHYSICAL MEDIUM

2.1. The Neutrino Helicity Flip Amplitude

The neutrino helicity flip occurs during the neu�trino scattering via an intermediate photon (plasmon)by the electromagnetic current of plasma compo�nents—electrons, νLe– νRe–, protons, νLp νRp, etc. The total Lagrangian of the process consistsof two parts. The first part describes the interaction ofthe neutrino that has a magnetic moment with pho�

72


KUZNETSOV et al.

tons, while the second part describes the interaction ofthe plasma with photons:

(8)

where σαβ = (1/2)(γαγβ – γβγα); γα, γβ are Dirac gammamatrices; F αβ is the photon electromagnetic field ten�sor; Jα = –( γαe) + ( γαp) + … is the electromagneticcurrent in the general case formed by the various com�ponents of the medium, i.e., free electrons andpositrons, protons, free ions, etc.; and Aα is the 4�potential of the photon electromagnetic field. Theneutrino magnetic moment is generally a matrix

≡ that contains both diagonal and transition mag�netic moments, where νi and νj are the states of a neu�trino with a specific mass. The neutrino states νl withspecific flavors l being produced in weak processes aresuperpositions of states νi:

(9)

where Uli is the unitary Pontecorvo–Maki–Naka�gawa–Sakata leptonic mixing matrix. Below, for sim�plicity, we will consider the diagonal neutrino mag�netic moment μν. The extension to the general case of

the matrix of magnetic moments presents no diffi�culty and consists in the following: the magneticmoment in all of the succeeding expressions should beconsidered as an effective value. For example, for theprocesses with initial electron neutrinos, by μν weshould mean

(10)

and a similar quantity for initial muon and tau neutrinos.The neutrino helicity flip during the scattering by

plasma components is described by the Feynman dia�gram shown in Fig. 1.

The technique for calculating the probability of theneutrino helicity flip is quite standard. The invariant

� i2�� μν

ij νjσαβνi( )Fαβ

i j,

∑– eJαAα,–=

e p

μνiνj

μν

ij

νl Uli*νi,

i

∑=

μν

ij

μν μνeμν

i jUej

j

∑2

i

∑⎝ ⎠⎜ ⎟⎛ ⎞

1/2

,≡

amplitude of the neutrino scattering by the kth plasmacomponent can be written as

(11)

where is the Fourier transform of the current of theneutrino magnetic moment,

is the Fourier transform of the kth component ofthe electromagnetic plasma current, and Q = (q0, q) isthe transferred 4�momentum. The only fundamentalpeculiarity is the use of the photon propagator Gαβ(Q)with allowance made for the plasma polarizationeffects. We apply the direct method of allowance forthese effects by summing the Feynman diagrams forthe forward photon scattering by plasma particles.This summation leads to Dyson’s equation. As aresult, for the plasmon propagator in the domainwhere the photon polarization operator is real, weobtain

(12)

where Πt, l are the eigenvalues of the photon polariza�tion tensor Παβ for transverse and longitudinal plas�mons,

(13)

ραβ(t, l) are the corresponding density matrices:

(14)

(15)

(16)

uα is the plasma velocity 4�vector. The density matricesραβ(λ) with λ = t, l have the properties of projectionoperators:

(17)

In the domain where the eigenvalues Πt, l of thephoton polarization tensor acquire imaginary parts,we will write them as

(18)

where Rλ and Iλ are the real and imaginary parts con�taining the contributions from all components of theactive medium. To extract the imaginary parts It, l, itwill suffice to make the analytic continuation q0 q0 + i� corresponding to the retarded polarizationoperator. The contributions from individual plasmacomponents (electrons, protons, etc.) are summed:

(19)

� k( )ieμν j ν( )

α Gαβ Q( )J k( )

β,–=

j ν( )

α

j ν( )

α νR p '( )σμανL p( )[ ]Qμ,=

J k( )

β

Gαβ Q( )iραβ t( )

Q2 Πt–��

iραβ l( )

Q2 Πl–��,+=

Παβ Πtραβ t( )– Πlραβ l( ),–=

ραβ t( ) gαβQαQβ

Q2��–

LαLβ

L2��–⎝ ⎠

⎛ ⎞ ,–=

ραβ l( )LαLβ

L2��,–=

Lα Qα uQ( ) uαQ2,–=

ραν λ( )ρβ

ν λ '( ) δλλ' ραβ λ( ).–=

Πλ Rλ iIλ,+=

Rt l, Rt l,e( ) Rt l,

p( ) …, It l,+ + It l,e( ) It l,

p( ) …+ += =

νL νR

J

γ*

Fig. 1. Feynman diagram for the neutrino helicity flip dur�ing the scattering via an intermediate plasmon γ* by theelectromagnetic plasma current J.



2.2. Eigenvalues of the Photon Polarization Tensor in Plasma

The scattering by plasma electrons and protonsmakes a major contribution to the neutrino helicityflip in a supernova core. The real and imaginary partsof the proton contribution to the eigenvalues of thephoton polarization tensor can be extracted from [23]by neglecting the antiproton contribution in the fol�lowing form:

(20)

(21)

(22)

(23)

where ηp is the proton chemical potential, T is the

temperature, v = P/�, � = , P and � arethe proton energy and momentum, mp is the effectiveproton mass in the plasma dependent on the protonnumber density Np [24, p. 152], α is the fine�structure

constant, and Pmin = mp . Here, we intro�duced the dimensionless parameter x = q0/q for which

< 1 in the scattering channel under consideration.The proton chemical potential ηp can be deter�

mined from the plasma electroneutrality condition,which can be written by neglecting the ion contribu�tion (see below) as

(24)

A numerical analysis of Eq. (24) shows that the differ�ence ηp – mp (the so�called nonrelativistic protonchemical potential) is positive at temperatures T ≈ 30–60 MeV, being of the same order of magnitude as thetemperature. Thus, under supernova core conditions,both the approximation of a degenerate Fermi gas andthe approximation of a classical Boltzmann gas aregenerally inapplicable for protons. However, we veri�

Rtp( ) 4α

π�� PP2d

� � ηp–( )/T[ ]exp 1+{ }��

0

∞

∫=

+ x2 x 1 x2–( )2v

�� x v+x v–��ln+⎝ ⎠

⎛ ⎞ ,

Itp( )

2αx 1 x2–( ) PP2d� ηp–( )/T[ ]exp 1+

��,

Pmin

∞

∫–=

Rtp( ) 4α

π�� 1 x2–( ) PP2d

� � ηp–( )/T[ ]exp 1+{ }��

0

∞

∫=

× 1 1 x2–

v2 x2–

�� xv

�� x v+x v–��ln–+⎝ ⎠

⎛ ⎞ ,

Ilp( ) 2It

p( )–=

+ 2αmp2x

mp

T 1 x2–��

ηp

T��–

⎝ ⎠⎜ ⎟⎛ ⎞

exp 1+1–

,

P2 mp2+

x / 1 x2–

x

Np Neηe

3

3π2��≈ ≈ 1

π2�� PP2d

� ηp–( )/T[ ]exp 1+�� .

0

∞

∫=

fied by direct calculations that the observed quantitiescalculated in Section 3, such as the luminosity (55),are barely sensitive to the choice of an approximationfor the distribution of protons.

Under supernova core conditions, the electronfraction may be considered with a good accuracy as adense relativistic degenerate plasma (ηe � T � me).Indeed, according to present views, the electronchemical potential in a supernova core reaches 200–250 MeV, while the temperature is 20–25 MeV. Underthese conditions, the real and imaginary parts (19) ofthe electron contributions to the photon polarizationfunctions can be obtained from Eqs. (20)–(23) by sub�stituting ηp ηe, mp me and by passing to thelimit me 0, T 0 in the form

(25)

(26)

(27)

(28)

where mγ is the so�called thermal photon mass,

(29)

As an illustration, Fig. 2 presents the electron andproton contributions to the eigenvalues Πl, t for longi�tudinal and transverse plasmons at temperature T =25 MeV and electron chemical potential ηe =250 MeV. The importance of allowing for the protoncontribution is obvious.

Along with the electrons and protons, some frac�tion Yi of free ions with charge eZi, number density ni,and mass mi can generally be also present in theplasma. This fraction may be considered with a goodaccuracy as a classical Boltzmann gas. The real andimaginary parts of the corresponding polarizationfunctions for a longitudinal plasmon are

(30)

where x0 = and we introduced the function

(31)

Rte( ) mγ

2 x2 x 1 x2–( )2

�� 1 x+1 x–��ln+⎝ ⎠

⎛ ⎞ ,=

Ite( ) π

2��mγ

2x 1 x2–( ),–=

Rle( ) 2mγ

2 1 x2–( ) 1 x2�� 1 x+

1 x–��ln–⎝ ⎠

⎛ ⎞ ,=

Ile( ) πmγ

2x 1 x2–( ),=

mγ

2 2απ

��ηe2.≈

Rli( ) 4πα

Zi2ni

T�� 1 φ x

x0

��⎝ ⎠⎛ ⎞– ,=

Ili( )

8π3/2αZi2ni

1x0q��

q0

2T��⎝ ⎠⎛ ⎞sinh=

× q2

8miT��⎝ ⎠⎛ ⎞ x2

x02

��–⎝ ⎠⎜ ⎟⎛ ⎞

,expexp

2T/mi

φ y( ) 2

π�� y 3 u 1 u+

1 u–�� y2u2–( )expln u.d

0

∞

∫=

74


KUZNETSOV et al.

As we see from Eqs. (30), the function can dif�fer markedly from zero only in a narrow range of thevariable x = q0/q or, more specifically, x � x0 ~

� 1.

On the other hand, the functions and for a

transverse plasmon are of the order of α ni/mi, sothat they are suppressed by the large ion mass in thedenominator. Therefore, only the contribution fromthe neutrino scattering by free ions via a longitudinalplasmon (λ = l) can be significant.

The ion contribution (30) will enter into any

observed characteristics with the factor Yi and it isnegligible under supernova core conditions, because Yi

is small. However, it can be significant in the super�nova envelope, which presumably can be saturated byiron�group metals.

2.3. The Probability for the Creation of Right�Handed Neutrinos

The quantity that defines the probability (E ')

for the creation of a right�handed neutrino with a fixedenergy E ' summed over all states of the initial left�handed neutrinos is of physical interest. This function

Ili( )

T/mi

Rti( ) It

i( )

Zi2

Zi2

ΓνR

can be obtained by integrating the square of amplitude(11) over the states of the initial left�handed neutrinosand over the states of the initial and final plasma parti�

cles that form the electromagnetic current :

(32)

Here, p = (E, p) and p' = (E ', p') are the 4�momenta ofthe initial and final neutrinos, � = (�, P) and �' =(�', P') are the 4�momenta of the initial and final

plasma particles; denotes summation over the

spins of these particles, the index k = e, p, i, … corre�sponds to the type of plasma particles (electrons, pro�tons, free ions, etc.) with a distribution function fk(�)that can be both fermion (the upper sign in the expres�sion in square brackets) and boson (the lower sign)one; fν(E) = {exp[(E – ηn)/T] + 1}–1 is the Fermi–Dirac distribution function for the initial left�handedneutrinos in the plasma rest frame, ην is the neutrino

J k( )

β

ΓνRE '( ) 1

16 2π( )5E '�� k( ) 2

s s ',

∑k

∑∫=

× δ 4( ) p ' � ' p– �–+( )dP��fk �( )

× dP'� '�� 1 fk � '( )+−[ ]dp

E��fν E( ).

s s ',∑

1.0−0.5

100

300

0 0.5x

−1.0

(c)200

0

Rt, MeV2

1.0−0.5

0

1000

0 0.5x

−1.0

(a)

500

−500

Rl, MeV2

1.0−0.5

50

150

0 0.5x

−1.0

(d)100

0

It, MeV2

1.0−0.5

−500

500

0 0.5x

−1.0

(b)

0

−1000

Il, MeV2

1000

−50

−100

−150

Fig. 2. Electron (dotted lines) and proton (dashed lines) contributions for the real (a) and imaginary (b) parts of Πl as well as forthe real (c) and imaginary (d) parts of Πt.



chemical potential, δ(4) is the 4�dimensional deltafunction.

In Eq. (32), it is convenient to pass from the inte�gration over the initial neutrino momentum p to theintegration over the virtual plasmon momentum p – p' =Q = (q0, q), ≡ q. This passage is defined by therelation

where θ(x) is the Heaviside step function. Substitutingthe square of amplitude (11) into Eq. (32) yields

(33)

where we introduced the tensor integral

(34)

It is convenient to calculate the tensor Tαβ using itscovariant properties. This requires writing the distri�bution function fk(�) in an arbitrary frame of refer�ence:

(35)

where uα is the plasma velocity 4�vector. This vectorand the 4�vector Qα are the “building material” for theformation of the tensor Tαβ. Since the electromagnetic

current is real, the tensor Tαβ is symmetric. Inaddition, it is orthogonal to the 4�vector Qα in view ofthe electromagnetic current conservation. It is easy tosee that there exist only two independent structureswith such properties—these are the density matrices(14) and (15). Thus, we can write

(36)

Since the tensors ραβ(t) and ραβ(l) are orthogonal,see (17), we obtain

(37)

q

dpE�� fν E( ) 2π

E '��qdqdq0θ Q2–( )=

× θ 2E ' q0 q–+( ) fν E ' q0+( ),

ΓνRE '( )

μν

2

8π2E '2

�� q0 q q fν E ' q0+( )d

q0

2E ' q0+

∫d

E '–

∞

∫=

× j ν( )

α j ν( )

α '* ραβ λ( )ρα 'β ' λ '( )

Q2 Πλ–( ) Q2 Πλ '*–( )��Tββ '

,

λ λ ',

∑

Tαβ e2

32π2�� J k( )

α J k( )

β* Φ,d∫s s ',

∑k

∑=

dΦ dPdP '��'

��fk �( ) 1 fk � '( )+−[ ]δ 4( ) � ' �– Q–( ).=

fk �( ) � u⋅ η–T

��⎝ ⎠⎛ ⎞ 1±exp

1–

,=

J k( )

α

Tαβ � t( )ραβ t( ) � l( )ραβ l( ).+=

� t( ) 12��Tαβραβ t( )=

= e2

64π2��ραβ t( ) J k( )

α J k( )

β* Φ,d∫s s ',

∑k

∑

(38)

It can be shown that integrals (37) and (38) definethe probabilities for the absorption (at q0 > 0) and cre�ation (at q0 < 0) of a plasmon by plasma particles.Indeed, to be specific, let us consider the probabilityfor the absorption of a transverse plasmon by thatplasma particles that form the electromagnetic current

. The amplitude of the process is

(39)

where εα(t) is a unit polarization 4�vector. Performingstandard calculations, we find the probability for plas�mon absorption by all plasma components,

(40)

where the summation is over all k types of plasma par�ticles and over the polarizations of all the particlesinvolved in the process, the index τ = 1, 2 numbers theplasmon polarizations, the indices s and s' were intro�duced for the plasma particle spin states.

Substituting the amplitude (39) into (40),

(41)

where

and comparing the derived expression with (37), wecan find the quantity

(42)

Using the well�known relation [25] between thetransverse plasmon absorption probability and theimaginary part It of the eigenvalue Πt of the photonpolarization tensor Παβ,

(43)

we will express �(t) in terms of It:

(44)

where fγ(q0) = [exp(q0/T) – 1]–1 is the Bose–Einsteindistribution function for photons. This relationderived in the case of q0 > 0 is also valid for the case ofq0 < 0, which corresponds to the creation of a trans�verse plasmon with an energy ω = –q0 > 0. Here, therelation between the imaginary part It and the trans�

� l( )Tαβραβ l( )=

= e2

32π2��ραβ l( ) J k( )

α J k( )

β* Φ.d∫s s ',

∑k

∑

J k( )

β

� k( )t( ) eεα t( )J k( )

α,–=

Γ t( )abs 1

32π2q0

��12�� k( )

t( )2Φ,d∫

s s ',

∑k

∑τ

∑=

Γ t( )abs e2

64π2q0

��ραβ t( ) J k( )

α J k( )

β* Φ,d∫s s ',

∑k

∑=

ραβ t( ) εατ t( )εβ

τ t( ),τ 1=

2

∑=

� t( )q0Γ t( )

abs.=

It q0( ) q0 1 q0/T–( )exp–[ ]Γ t( )abs

,–=

� t( ) It

1 q0/T–( )exp–��– It 1 fγ q0( )+[ ],–= =

76


KUZNETSOV et al.

verse plasmon creation probability should beapplied:

(45)

The oddness of the function It is also important,

(46)

and this peculiarity originates from the properties ofthe retarded polarization operator.

It is easy to verify that Eq. (44) is also valid for alongitudinal plasmon. It should be remembered thatραβ(l) = –εα(l)εβ(l) and

(47)

Eventually, we obtain the tensor Tαβ in the form

(48)

Substituting (48) into (33), using the orthogonalityof the tensors ραβ(t) and ραβ(l) (17), and taking intoaccount the expressions for the convolutions of theneutrino current with these tensors,

we ultimately obtain the probability for the creation ofright�handed neutrinos

(49)

Here, we introduced the spectral densities of the plas�mon

(50)

that are defined by the eigenvalues (18) of the photonpolarization tensor (13).

Formula (49) presents our main result. Note that itis in agreement, to the notation, with the probabilityobtained in [12] by analyzing the retarded self�energyoperator of a right�handed neutrino. However,extracting only the electron contribution from ourgeneral expression, we have a result that is twice thatobtained in [20, 21]. It can be seen that in these papersthe error is contained directly in the first formula that

Γ t( )cr

It ω( ) ω ω/T( )exp 1–[ ]Γ t( )cr

.–=

It q0–( ) It q0( ),–=

Il q0( ) q0 1 q0/T–( )exp–[ ]Γ l( )abs

.=

Tαβ Itpαβ t( )– Ilρ

αβ l( )–[ ] 1 fγ q0( )+[ ].=

j ν( )

α j ν( )

β*ραβ t( ) Q4 2E ' q0+( )2

q2�� 1– ,=

j ν( )

α j ν( )

β*ραβ l( ) Q4 2E ' q0+( )2

q2��,–=

ΓνRE '( )

μν

2

16π2E '2

�� q0 q3 qd

q0

2E ' q0+

∫d

E '–

∞

∫=

× fν E ' q0+( ) 2E ' q0+( )2 1q0

2

q2��–

⎝ ⎠⎜ ⎟⎛ ⎞

2

× 1 fγ q0( )+[ ] 1 q2

2E ' q0+( )2��–

⎝ ⎠⎜ ⎟⎛ ⎞

t l– .ζρ ζρ

λ

2Iλ

Q2 Rλ–( )2

Iλ2+

��,–=ζρ

defines the probability for the creation of right�handedneutrinos.

Our formula (49) derived for the interaction ofneutrinos with virtual photons actually has a moregeneral meaning and can be used for the neutrino–photon processes in any optically active medium. It isonly necessary to determine the photon spectral den�

sity functions . For example, in a medium where

It 0 in the space�like region Q2 < 0, correspondingto refractive indices n > 1, the spectral density functiontransforms into a delta function and we can reproducethe result of [26], which is devoted to the investigationof the so�called neutrino Cherenkov radiation, wherea neutrino in the medium emits a transverse photon.

2.4. The Contributions from Plasma Components to the Neutrino Scattering Processes

As was noted above, the neutrino helicity flip pro�cesses are analyzed by taking into account the neu�trino scattering by various plasma components: elec�trons, protons, free ions, etc. Since these processes areincoherent and the probabilities of various channelsare added, it makes sense to consider the separate con�tribution from the neutrino scattering by electrons tothe probability for the creation of right�handed neutri�nos. This means that only the electron contribution tothe function Iλ in the numerator of Eq. (50) should betaken into account. Note, however, that the functionsRλ and Iλ in the denominator of Eq. (50) will containthe contributions from all plasma components. Here,our result for the neutrino scattering by electrons dif�fers from the result of [20, 21], where only the electroncontribution was taken into account not only in thenumerator but also in the denominator of the expres�sion for the plasmon spectral densities, which isimproper.

As we see from our analysis performed in Section 2.2,the electron and proton contributions to the imaginaryparts Iλ of the eigenvalues Πλ of the photon polariza�tion tensor are of the same order of magnitude andhave the same sign both at λ = t and at λ = l, seeFigs. 2b and 2d. This circumstance per se must lead toa decrease in the electron contribution to the function

(E '). On the other hand, we see from Fig. 2a that

the electron and proton contributions to the real partRl of the eigenvalue Πl are of the same order of magni�tude but have opposite signs in the domain where theimaginary part of the electron contribution in thenumerator of Eq. (50) is relatively large. As a result,the contribution from the neutrino scattering by elec�trons to the probability for the creation of right�handed neutrinos that we obtained is close to the resultof [20, 21], except for the factor of 2 mentioned above.

The contribution from the neutrino scattering byprotons to the probability for the creation of right�handed neutrinos can be investigated in a similar way.

λζρ

ΓνR



In this case, only the proton contribution to the func�tions Iλ (21), (23) in the numerator of Eq. (50) shouldbe taken.

The results of our numerical analysis of the individ�ual contributions from the neutrino scattering by elec�trons and protons and the total probability for the cre�ation of neutrinos νR under typical supernova coreconditions are presented in Fig. 3.

The dimensionless width �(E ') shown in the graphis defined by the expression

(51)

For comparison, the result of [21] is also shown inFig. 3, which illustrates a gross underestimation of theneutrino helicity flip probability assumed by theseauthors.

We also considered the contribution from the neu�trino scattering by free ions to the probability for thecreation of neutrinos νR. It should be noted that, inthis case, the longitudinal virtual plasmons make amajor contribution to the νR creation probability. As

we see from Eqs. (30), the function is nonzero onlyin a narrow range Δx of the variable x = q0/q or, more

specifically, Δx ~ � 1. This makes it possible tocalculate the ion contribution to the probability for thecreation of neutrinos νR analytically and to obtain therelation

(52)

where mD has the meaning of a reciprocal Debye

screening length, = (q0 = 0), where the

summation is over all plasma components. Interest�ingly, Eq. (52) derived in the heavy�ion approxima�tion, describes satisfactorily the proton contribution.

Using the derived function (E '), we can calcu�

late the total number of right�handed neutrinos emit�ted per 1 MeV of the energy scale in a unit time from aunit volume, i.e., the energy spectrum of right�handedneutrinos:

(53)

The energy spectrum (53) of right�handed neutri�nos can be used to investigate the possible mechanismsof energy transfer by these neutrinos to the outer layersof a supernova envelope. For example, some of theright�handed neutrinos can be inversely convertedinto left�handed ones followed by their absorption.These processes were suggested by Dar [27] and theninvestigated in [28–31] as one of the presumed mech�anisms for stimulating a damped shock during a super�nova explosion.

ΓνRE '( ) μν

2ηe3� E '( ).=

Ili( )

T/mi

ΔΓνR

i( ) E '( ) μν

2αZi2ni fν E '( )=

×4E '

2 mD2+

mD2

�� 4E ' 2

4E '2 mD

2+��–ln

⎝ ⎠⎜ ⎟⎛ ⎞

,

mD2 Rl

k( )

k∑

ΓνR

dnνR

dE '�� E '

2

2π2��ΓνR

E '( ).=

3. CONSTRAINTS ON THE NEUTRINO MAGNETIC MOMENT

The formulas derived above can be used to placeupper limits on the neutrino magnetic moment bycomparing the supernova core luminosity calculatedfrom the energy spectrum (53) of right�handed neutri�nos with the left�handed neutrino luminosity ~

1052–1053 erg/s [24, 32].

In the next subsections, we make a more reliableestimate of the neutrino magnetic moment than hasbeen done previously by taking into account the radialdistributions and time evolution of physical parame�ters from realistic supernova core models. We obtainedupper limits on the combination of the effective mag�netic moments of electron, muon, and tau neutrinos.The condition for a sufficiently small influence of theright�handed neutrino emission within 1–2 s after col�lapse on the subsequent supernova explosion stage, theso�called Kelvin–Helmholtz stage, provides a basisfor these constraints. The models are considered inreverse chronological order.

3.1. A Model for the Collapse of a Supernova with an O–Ne–Mg Core

At present, there exist several groups of astrophysi�cists concerned with supernova explosion simulations.At our request, H.�T. Janka (Garching, Germany)kindly provided the as yet unpublished results [33] ofrecent simulations of oxygen–neon–magnesiumsupernova core collapse to us. This model by Jankawith coauthors is a continuation of the simulationsdescribed in [34, 35]. The result of a successful explo�sion for this model has recently been confirmed inde�

LνL

50

1

3

100 150E ', MeV

0

2

R × 104

200

4

5

Fig. 3. Contributions from electrons (dashed line) andprotons (dash–dotted line) to the dimensionless creationwidth �(E ') of a right�handed neutrino and total width(solid line) for plasma temperature T = 25 MeV and chem�ical potentials of electrons ηe = 250 MeV and neutrinos

= 100 MeV. The dotted line indicates the result of [21].ηνe

78


KUZNETSOV et al.

pendently by a supernova simulation group working atthe Universities of Arizona and Princeton (USA)[36, 37], which reached similar conclusions.

It should be emphasized that this O–Ne–Mg corecollapse model (for an initial stellar mass of 8.8M�) isinapplicable directly to SN1987A, whose mass beforeits collapse is estimated to be (15–20)M� and, accord�ing to the general theory of stellar evolution, the pro�genitor of SN 1987A had a collapsing core composedof the elements up to iron. However, as will be seenfrom the subsequent discussion, the constraint on theneutrino magnetic moment obtained in this model isclose to the constraints obtained by analyzing othermodels with Fe�core collapse.

For our purposes, we should write an equation forthe spectral density of the supernova core luminositythrough right�handed neutrinos where the integrationshould be performed over the volume of the neutrino�emitting region to obtain the spectral density of thesupernova core luminosity via right�handed neutrinos:

(54)

As an illustration, Fig. 4 shows this luminosity spec�trum for electron chemical potential ηe = 250 MeVand for several plasma temperatures.

Using here the quantities defined in Eqs. (49) and(50) and in the corresponding formulas from Section 2.2of this paper, we take into account their dependenceon radius R and time t. The full set of the distributionsof parameters [33] used in our estimations includes theprofiles of density ρ, temperature T, electron fractionYe, electron neutrino, , and electron antineutrino,

dLνR

dE '�� V E '

3

2π2��ΓνR

E '( ).d∫=

Yνe

, fractions, and fractions for the remaining

types of neutrinos and antineutrinos (νx = νμ, τ, ),which are considered to be identical. The time evolu�tion of the distributions of parameters was calculatedin [33] within approximately 2 s after collapse. As anillustration, Fig. 5 presents the radial distributionsinside the supernova core from 0 to 20 km at time t =1.0 s after collapse for the temperature [33], the chem�ical potentials of electrons ηe and electron neutrinos

(calculated from data in [33]), and the nonrelativ�

istic chemical potential of protons = ηp – thatdefines the degeneracy of protons (the calculation isbased on data from [33] using the effective nucleonmass mN in plasma [24, p. 152]).

To analyze the contribution from the right�handedneutrino emission to the total energy loss by the super�nova, we also use the time evolution of the total lumi�nosity of all types of left�handed neutrinos [33] pre�sented in Fig. 6.

Integrating Eq. (54) over the neutrino energy, weobtain the time evolution of the right�handed neutrinoluminosity:

(55)

Right�handed neutrinos are a new cooling agent thatmust compete with the energy losses through activetypes of neutrinos in order to affect significantly thecooling rate of the stellar core. Thus, the observedduration of the neutrino signal from SN1987A indi�cates that the additional energy losses through right�handed neutrinos must be limited from above at leastby the losses through left�handed neutrinos,

(56)

We will assume that this estimate is also applicable tothe O–Ne–Mg core collapse model under consider�ation. Within 2 s after collapse, we obtain a time�dependent upper limit on the combination of theeffective magnetic moments of electron, muon, andtau neutrinos from Eqs. (55) and (56). If we assume,for simplicity, that these effective magnetic momentsare equal, then we will obtain the time evolution of theupper limit on some flavor�averaged neutrino mag�netic moment . This dependence is shown in Fig. 7,

where = ( /μB) × 1012.

As we see from Fig. 7, the averaged upper boundtends to a certain value that gives the limit

(57)

YνeYνx

νμ τ,

ηνe

ηp* mN

LνRt( ) 1

2π2�� V E'E '

3ΓνRE '( ).d

0

∫d∫=

LνRLνL

.<

μν

μ12 μν

μν 2.4 10 12– μB.×<

100

1

3

200 300E ', MeV

0

2

400

4

5

dLνR

dE '�� 10

48erg s

–1MeV

–1,

Fig. 4. Energy distributions of the right�handed neutrinoluminosity for plasma temperatures T = 35 MeV (solidline), T = 25 MeV (dashed line), T = 15 MeV (dash–dot�ted line), T = 5 MeV (dotted line) and neutrino magneticmoment μν = 3 × 10–13

μB.



Generally, the combined constraint on the effectivemagnetic moments of electron, muon, and tau neutri�nos has the form

(58)

where the effective magnetic moments are defined byEq. (10). This limit is less stringent than the constraintobtained in the model of a homogeneous sphere forthe supernova core [22], but it, of course, is more reli�

μνe

2 0.71 μνμ

2 μντ

2+( )+[ ]1/2

3.7 10 12– μB,×<

able. In addition, it places an upper limit on the effec�tive magnetic moments of muon and tau neutrinos.

3.2. Early Supernova Explosion Models

We performed such an estimation procedure usingthe model data from [38] for two�dimensional hydro�dynamic simulations of supernova core collapse at astellar mass of 15M�. The radial distributions ofparameters at times t = 0.2, 0.4, 0.6, and 0.8 s after col�lapse in the s15Gio_32.a model were taken fromFig. 40 in [38]. In addition, the electron neutrino frac�tion was estimated as ≈ (1/5)Ye. When calculating

the luminosity of right�handed neutrinos with suchparameters and using the constraint (56), where thetotal luminosity through active types of neutrinos

in this model can be taken from Fig. 42 in [38], we findthat the upper limit on the flavor�averaged neutrinomagnetic moment also changes with time, as in the

previous case. The time�averaged upper bound for corresponding to the interval 0.4–0.8 s is

(59)

and is close to the limit (57).

The thermal and chemical evolution over theKelvin–Helmholtz formation stage of a neutron starwas investigated in [39]. Using the data presented inFigs. 9 and 14 from [39], we obtained a time�averagedupper bound for within 1–10 s of evolution aftercollapse in the form

(60)

We also used the results of [40], where the numeri�cal simulations of the neutrino cooling of a recentlyformed, lepton�enriched neutron star are presented.Based on the data presented in Figs. 3–9 for the SBHmodel (a hot star with a relatively low baryon mass)

Yνe

LνL

μν

μν

μν 2.7 10 12– μB×<

μν

μν 1.2 10 12– μB.×<

5

50

150

10 15R, km

0

100

20

η, MeV

(b)

5

−50

50

10 15R, km

0

0

20

100

(c)

−100

ηp*, MeV

200

250

5

10

30

10 15R, km

0

20

20

T, MeV

(a)40

Fig. 5. Radial distributions of the temperature (a) [33] andthe chemical potentials of electrons (b, solid line), electronneutrinos (b, dashed line), and nonrelativistic chemicalpotential of protons = ηp – (c) inside the supernova

core at time t = 1.0 s after collapse.

ηp* mN

0.5

0.5

1.5

1.0 1.5t, s

0

1.0

2.0

2.0

LνL

tot10

53erg/s,

Fig. 6. Time evolution of the total luminosity of neutrinosof all active types [33].

80


KUZNETSOV et al.

[40], we estimated a time�averaged upper limit on for the interval 0.5–5 s after collapse in the form

(61)

Summarizing all our estimates (57), (59), (60), and(61), we see that the upper limit on the flavor�andtime�averaged neutrino magnetic moment at theKelving–Helmholtz stage of a supernova explosioncan be presented in the form

(62)

depending on the explosion model.

4. CONCLUSIONS

Here, we studied in detail the Dirac neutrino helic�ity flip under supernova core conditions for which theleft�handed neutrinos produced through the interac�tion of the magnetic moments with plasma electronsant protons can be converted into right�handed neu�trinos sterile with respect to weak interactions. Whencalculating the probability for the conversion neutrinoscattering by plasma components, we took intoaccount the polarization effects in the photon propa�gator attributable to the contributions from both elec�trons and protons. We calculated the probability

(E ') for the creation of right�handed neutrinos

with a fixed energy E ', their energy spectrum, andluminosity. Instead of the model of a homogeneoussphere for the supernova core, which was used in pre�vious studies, we considered more realistic modelswith radial distributions and time evolution of physicalparameters in the supernova core. We obtained explo�sion�model�dependent upper limits on the Dirac neu�trino magnetic moment averaged over flavors and time

μν

μν 1.1 1012– μB.×<

μν 1.1–2.7( ) 10 12– μB×<

ΓνR

on condition that the influence of the right�handedneutrino emission on the total cooling time scale waslimited (see inequality (56)).

(63)

ACKNOWLEDGMENTS

We thank Hans�Thomas Janka, Lorentz Hüde�pohl, and Bernard Müller, who provided detailed dataon the radial distributions and time evolution of phys�ical parameters in a supernova core obtained in theirmodel of supernova explosion and protoneutron starcooling. We are grateful to V.A. Rubakov, G. Raffelt,and O. Lychkovsky for useful remarks. This study wascarried out within the framework of the “Scientificand Scientific–Educational Personnel of InnovationRussia” Federal Goal�Oriented Program for 2009–2013 (State contract no. P2323) under partial finan�cial support by the Ministry of Education and Scienceof the Russian Federation according to the “Develop�ment of Scientific Potential of Higher School” Pro�gram (project no. 2.1.1/510).

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Translated by V. Astakhov

Plasma-induced neutrino helicity flip in a supernova core and a constraint on the Dirac neutrino...

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